FORMAL REALI/.AHII.ITY THIOOKY — I 245 



10.35. The abstract scalar product of 10.11 turned out in the end to be 

 no more than the concrete one of 7.0 when we restrict our attention to 

 //-tuples derived from vectors by the use of coordinate frames. In a 

 similar way, it is not iiard to show that there always exists a coordinate 

 frame in which the abstract conjugation now introduced has the form 

 of 7.2. This will be done in the Appendix (20.2). 



10.36* Our need for writing out the components of vectors has now 

 almost vanished. Henceforth we shall use subscripts to denote particular 

 vectors, e.g. Vi , rather than components. 



10.4* A vector will be called real if it is equal to its own (!onjugate. 

 A manifold will be called real if it contains with each vector also the 

 conjugate of that vector. V and K are then real. A basis will be called 

 real if it is made up of real vectors, and a frame will be called real if its 

 bases are real. Any frame in terms of which our conjugation operation 

 takes the form of 7.2 is real by definition because its basis vectors in 

 that frame have components which are or 1. The vector is real, 

 similarly. 



10.41* The basis dual to a real basis is real, for if 



\Vr, ks) = Ors , 



then by (10) of 10.3 and the hypothesis that Vr = Vr , we have 



(iV , ks) = 8rs = 8rs 



so the ks satisfy the same equations as the k. The uniqueness of the 

 basis dual to i'l , • • • , Vr then proves that ks = ks , I < s < n. 



10.42* Any vector v can be written 



where Vi and ro are real. Namely 



vi = 2 (^ + ^^■ 



V2 = -^.{v - v). 



10.5* It is shown in Halmos'', par. 34, that if reV, AeK are represented 

 by [i'], [k] in some coordinate frame, and by [r]i , [A']i in some other frame, 

 then there is a nonsingular matrix [IT], which (a) depends only upon the 



* Technical paragraph as explained in Section 2.91. 



